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MAT485-2003Spring

# MAT485-2003Spring - MAT 485(002 — FINAL EXAM Name May 6...

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Unformatted text preview: MAT 485 (002) — FINAL EXAM Name : May 6, 2003 Time : 2 hours. Total Marks: 100 Instructions 1. Attempt all questions. 2. Show all your steps, answers without justiﬁcation will not receive full credit. A mere ’yes’ or ’no’ is not acceptable. 3. This exam contains IX problems and 13 pages including a cover sheet. Make sure you have all the sheets. DO NOT WRITE BELOW THIS LINE I. II. III. IV. V. VI. VII. VIII. IX. TOTAL MAT 485 (002) - FINAL EXAM I. Solve the following system of equations using Gauss or Gauss- Jordan elimination. Show all the elementary row operations you are using. (10 marks) x+y=3 y+z=4 x+z=5 MAT 485 (002) - FINAL EXAM 3 Ila. Are the vectors (2,—4, —6), (—3,5,7),(—4, 5,6) linearly inde— pendent in R3? Justify your answer. (6 marks) 11 b. Are the functions f1(t) = 1,f2(t) : e‘t, f3(t) = t6_t linearly independent? Justify your answer. (6 marks) MAT 485 (002) - FINAL EXAM III. Solve the following initial value problem by separating the vari- ables y3y' + \$3 = 0, y(0) =1 (6 marks) MAT 485 (002) — FINAL EXAM 5 IV. Consider the differential equation (cosy + ycosm)dx + (sinx — xsiny)dy 2 0. (a) Is the equation exact? Justify your answer. (4 marks) (b) Find the general solution to the above differential equa— tion. (6 marks) MAT 485 (002) — FINAL EXAM V. A tank contains 200 gallons of fresh water. A solution contain— ing 2 lb/gal of salt is pumped into the tank at 5 gal/min. The mixture is stirred constantly and ﬂows out at the same rate of 5gal/ min. (a) What is the actual amount of salt y(t) at time t? (b) How much salt is in the tank after 10 minutes? (12 marks) MAT 485 (002) - FINAL EXAM Extra page for problem V. MAT 485 (002) - FINAL EXAM VI. Solve the IVP for the following Euler—Cauchy equation: 9321/” — 123/ + 5y = 0, y(1) = -2, y’(1)= 0 (10 marks) MAT 485 (002) - FINAL EXAM 9 VII. Solve the following IVP. y” + 3y’ + 2y = 36”, y(0) = 0, y’(0) = 0 (15 marks) 10 MAT 485 (002) - FINAL EXAM Extra page for problem VII. MAT 485 (002) - FINAL EXAM 11 VIII. Find a solution to the following system of linear differential equations: yi = 61/1 — 312 y; = 5y1 + 4y2 (15 marks) 12 MAT 485 (002) — FINAL EXAM Extra page for problem VIII. MAT 485 (002) - FINAL EXAM 13 IX. Find a power series solution in powers of :1: to the following differential equation (Show the details of your work): (1 - 332W 2 2w (10 marks) ...
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MAT485-2003Spring - MAT 485(002 — FINAL EXAM Name May 6...

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